Printing of ultrathin layers of polymeric and colloidal inks is critical for the manufacturing of electronics on nonconventional substrates such as paper and polymer films. Recently, we found that nanoporous stamps overcome key limitations of traditional polymer stamps in flexographic printing, namely, enabling the printing of ultrathin nanoparticle films with micron-scale lateral precision. Here, we study the dynamics of liquid transfer between nanoporous stamps and solid substrates. The stamps comprise forests of polymer-coated carbon nanotubes, and the surface mechanics and wettability of the stamps are engineered to imbibe colloidal inks and transfer the ink upon contact with the target substrate. By high-speed imaging during printing, we observe the dynamics of liquid spreading, which is mediated by progressing contact between the nanostructured stamp surface and by the substrate and imbibition within the stamp-substrate gap. From the final contact area, the volume of ink transfer is mediated by rupture of a capillary bridge; and, after rupture, liquid spreads to fill the area defined by a precursor film matching the stamp geometry with high precision. Via modeling of the liquid dynamics, and comparison with data, we elucidate the scale- and rate-limiting aspects of the process. Specifically, we find that the printed ink volume and resulting layer thickness are independent of contact pressure; and that printed layer thickness decreases with retraction speed. Under these conditions, nanoparticle films with controlled thickness in the <100 nm regime can be printed using nanoporous stamp flexography, at speeds commensurate with industrial printing equipment.